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The symbol := used in this assignment is called the assignment operation. 6 Set-Theoretic Operations We've caused ourselves extra effort by requiring that the assignment operation allow a redefinition of all the attribute names in the heading of the original table. We will see the value of this shortly. For now, we note that a redefinition of attributes is not always necessary, and when all attributes are identical between the two tables, Bi = Ai for all i, 1 < i < n, we refer to S as an alias of the table R and simply write S:=R ■ Note that the table R on the right can result from an evaluation of a relational algebra expression, and thus gives us an opportunity to "save" intermediate results of evaluation, much as we do with assignment statements used in programming languages.

A key for the table T is a set of attributes, K = Aj . . Aj , with two properties: [1] If u, v are distinct tuples of T, then by designer intention u[K] Φ v[K]; that is, there will always exist at least one column, A{ , in the set of columns K such that u[A; ] Φ v[A: ]. m [2] N o proper subset H of K has property 1. m Property 1 is just a mathematical way of saying that the values any row u takes on for the set of attributes K are unique. , u[K] = v[K]), they are really one row after all (u = v).

A condition of selection determines, for each given tuple of the table, whether that tuple is qualified to remain in the set of selected rows making up the answer. The form a condition C can take is defined recursively as follows. 53 Chapter 2 54 THE RELATIONAL MODEL [11 C can be any comparison of the form A; oc Aj, or A; oc a, where A; and Aj are attributes of S having the same domain, a is a constant from Dom(Ai), and , =, <=, >=, and <>. ) The table S where A^ oc Aj consists of all rows t of T with the property that t[AJ oc t[Aj]; the table S where Aj oc a contains all rows t of S with the property t[A;] oc a.